Initial pre-algebras as a generalization of dendriform algebras

We continue the study of \emph{initial dialgebras} defined in~\cite{DMS2026}.

For a binary operad $\Var$ we define the class of initial pre-$\Var$-algebras and the corresponding operad $\pre\Var^{\I}$ in such a way that \[ (\di\Var^{\I})^{!}=(\pre(\Var^{!}))^{\I} \] in the case when $\Var $ is quadratic.

We propose an intuitive algorithm for finding the defining relations of the operad $\pre\Var^{\I}$ in the case when $\Var$ is a binary quadratic operad.

We also study free initial pre-algebras in the associative and commutative settings.

For the nonsymmetric operad $\pre\As^{\I}$, we construct a Grobner--Shirshov basis in the free magma operad, describe a linear basis in terms of admissible decorated planar binary trees, and establish a bijection between these trees and certain combinatorial objects.